Shooting in the rain is less common than most other times, so your photos may be a little more unique.
Rain can be interesting and it can be easier to tell a story (especially if the story is that it’s raining). The lighting may be less interesting (lower contrast, less color) which can be tweaked and tweaked. Just drag around the curve and play with your sliders or bail out and convert to black & white.
But, you get wet, lots of wet (which can be a problem with some cameras).
Weather gear is available, though, Plastic bags with a lens port are easiest and cheapest, under $10 (Rainsleeve).
Or you can get a more durable fabric shell, around $50 (Shell). This is a good option if you think you’ll be out in the rain or snow a lot.
There are also waterproof point and shoots or mirrorless cameras which are good, but not of them all are DSLR quality and can be pricey for the better ones ($500++)
My all weather-super tough-always with me-pocket point and shoot is my Olympus Tough TG-5. It’s waterproof down to to 30 feet. I can drop it in a creek, fish it out and just keep shooting.
For the best quality photos a weather-sealed DSLR is an option (weather-sealed DSLRs). Weather sealing is great (very great) but it isn’t waterproof. Even if you have a weather-sealed puppy, the best bet is to also pick up some cheap rainsleeves and keep them handy.